Basic Mathematics
Angles and Measurements Straight Line Co-ordinate Geometry Distance Formula Section Formula Mid Point Formula Centroid Formula Incentre Formula Area Formulas Circle Ratios If a/b = c/d THEN ; Alternendo d/b = c/a Invertendo b/a = d/c Componendo (a+b)/b = (c+d)/d Dividendo ''' (a-b)/b = (c-d)/d '''Componendo - Dividendo (a+b)/(a-b) = (c+d)/(c-d) Theorem on Equal Ratios (a+c)/(b+d) = (a-c)/(b-d) = k Expansions (a+b)2 = a2 + b2 + 2ab (a-b)2 = a2 + b2 - 2ab a2 - b2 = (a-b)(a+b) (a+b)3 = a3 + 3a2b +3ab2 +b3 (a-b)3 = a3 - 3a2b + 3ab2 - b3 a3 + b3 = (a+b)(a2-ab+b2) a3 - b3 = (a - b)(a2+ab+b2) (a+b+c)2 = a2 + b2 +c2 + 2(ab+bc+ac) a2 + b2 = (a + b)2 - 2ab a2 + b2 = (a - b)2 + 2ab (a - b)2 = (a + b)2 - 4ab Indices Logarithms Identity f(x) = g(x) is an identity if both have the same value for all real values of x . Polynomials 1) First order polynomial : f(x) = 0 ; e.g. ax + b = 0 x = -b /a (only one solution) 2) Second Order Polynomial : ax2 + bx + c = 0 x = ± √(b2-4ac) / 2a (two solutions) 3) Third Degree Polynomial : Can be solved if the equation can be factorized into linear and quadratic sub-equations . 4) nth Degree Polynomial : has n solutions Exponential Equations e.g. af(x) = ag(x) , where a =/ 1 then f(x) = g(x) Equate the exponents to find the solution . If it cannot be put in the form , substitute another variable y so as to get a polynomial equation . Logarithmic Equations if logaf(x) = logag(x) ; then f(x) = g(x) . f(x) and g(x) should be positive . if the above form cannot be achieved , put a variable y , so as to get a polynomial equation . Functions Every function can be plotted on graph . Based on the behavior of graphs , we have an important section of Mathematics , called the Calculus , which studies the following :- # x and corresponding y co-ordinates # Continuity of graph of a function # Rate of change of y co-ordinate with respect to x co-ordinates # Revival of function from rate of change of y w.r.t. x . If a function is defined as max{ 2x + 5 , 4x + 7} ; it means that our function is the outer boundary amongst the to given functions . Some Important Functions A Function is an expression which defines the behaviour of numbers . Modulus Function : f(x) = |x-a| = x-a x>a = a-x x" , "<" instead of "=" is called an inequation . Laws of inequations 1) a + b > a + c ; then b > c 2) if a > b ; then ca > cb 3) if ab > ac ; then b > c ; if a > 0 4) if a > 0 , b > 0 ; then a+b > 0 , ab > 0 if a < 0 , b < 0 ; then a+b < 0 , ab > 0 if a<0 , b >0 ; then ab > 0 5) ax > ay ; then x > y if a > 1 6) logax > logay ; then x > y if a > 1 x < y if 0 < a < 1 for evaluating roots of an inequations , use basic rules discussed in the chapter earlier . The answer always lies as an inequation . Sign Scheme Tips and Tricks Category:Mathematics